Infinite-dimensional sphere

In algebraic topology, the infinite-dimensional sphere is the inductive limit of all spheres. Although no sphere is contractible, the infinite-dimensional sphere is contractible^[1][2] and hence appears as the total space of multiple universal principal bundles.

Definition

With the usual definition ${\displaystyle S^{n}=\{x\in \mathbb {R} ^{n+1}|\|x\|_{2}=1\}}$ of the sphere with the 2-norm, the canonical inclusion ${\displaystyle \mathbb {R} ^{n+1}\hookrightarrow \mathbb {R} ^{n+2},x\mapsto (x,0)}$ restricts to a canonical inclusion ${\displaystyle S^{n}\hookrightarrow S^{n+1}}$. Hence the spheres form an inductive system, whose inductive limit:^[3][4]

${\displaystyle S^{\infty }:=\lim _{n\rightarrow \infty }S^{n}}$

is the infinite-dimensional sphere.

Properties

The most important property of the infinite-dimensional sphere is, that it is contractible.^[1][2] Since the infinite-dimensional sphere inherits a CW structure from the spheres,^[3][5] Whitehead's theorem claims that it is sufficient to show that it is weakly contractible. Intuitively, the homotopy groups of the spheres disappear one by one, hence all do for the infinite-dimensional sphere. Concretely, any map ${\displaystyle S^{k}\rightarrow S^{\infty }}$, due to the compactness of the former sphere, factors over a canonical inclusion ${\displaystyle S^{n}\hookrightarrow S^{\infty }}$ with ${\displaystyle k<n}$ without loss of generality. Since ${\displaystyle \pi _{k}(S^{n})}$ is trivial, ${\displaystyle \pi _{k}(S^{\infty })}$ is also trivial.

Application

• ${\displaystyle S^{\infty }\twoheadrightarrow \mathbb {R} P^{\infty }}$ is the universal principal ${\displaystyle \operatorname {O} (1)}$-bundle, hence ${\displaystyle \operatorname {EO} (1)\cong S^{\infty }}$. The principal ${\displaystyle \operatorname {O} (1)}$-bundle ${\displaystyle S^{n}\twoheadrightarrow \mathbb {R} P^{n}}$ is then the canonical inclusion ${\displaystyle i\colon \mathbb {R} P^{n}\hookrightarrow \mathbb {R} P^{\infty }}$, hence ${\displaystyle S^{n}\cong i^{*}S^{\infty }}$.
• ${\displaystyle S^{\infty }\twoheadrightarrow \mathbb {C} P^{\infty }}$ is the universal principal U(1)-bundle, hence ${\displaystyle \operatorname {EU} (1)\cong \operatorname {ESO} (2)\cong S^{\infty }}$. The principal ${\displaystyle \operatorname {U} (1)}$-bundle ${\displaystyle S^{2n+1}\twoheadrightarrow \mathbb {C} P^{n}}$ is then the canonical inclusion ${\displaystyle j\colon \mathbb {C} P^{n}\hookrightarrow \mathbb {C} P^{\infty }}$, hence ${\displaystyle S^{2n+1}\cong j^{*}S^{\infty }}$.
• ${\displaystyle S^{\infty }\twoheadrightarrow \mathbb {H} P^{\infty }}$ is the universal principal SU(2)-bundle, hence ${\displaystyle \operatorname {ESU} (2)\cong \operatorname {ESp} (1)\cong S^{\infty }}$. The principal ${\displaystyle \operatorname {SU} (2)}$-bundle ${\displaystyle S^{4n+3}\twoheadrightarrow \mathbb {H} P^{n}}$ is then the canonical inclusion ${\displaystyle k\colon \mathbb {H} P^{n}\hookrightarrow \mathbb {H} P^{\infty }}$, hence ${\displaystyle S^{4n+3}\cong k^{*}S^{\infty }}$.

Literature

• Hatcher, Allen (2001). Algebraic Topology. Cambridge University Press. ISBN 978-0-521-79160-1.
• tom Dieck, Tammo (2008-09-01). Algebraic Topology (PDF). ISBN 978-3037190487.